where \(\varsigma _{j,i}\)  is the weight attached to the message \(p_{j,i}(x_j)\)  in the corresponding FG.
During the off-line training, since \(I_{i,j}(x_i, x_j)\)  is the interfering node that is irrelevant to the message \(p_{j,i}(x_j)\) , \(I_{i,j}(x_i, x_j)\varsigma _{j,i}\)  is treated as the trainable parameters in the neural network.
Training with the additional weights has shown to improve the sum-product algorithm's performance in a high SNR region [1]}.
However, merely tuning the additional weight \(\varsigma _{j,i}\)  attached to the message \(p_{j,i}(x_j)\)  will not change the connections in the FG. Therefore, the performance improvement by attaching tuneable weights to the NN is limited.
To this end, we propose to connect an NN FN \(\Phi (x_1,...,x_N)\)  to the VNs in the FG to compensate for the effects of the residual ISI responses and the correlation induced along the short cycles.
As shown in Fig. REF , different from the traditional FG, we nest an NN FN to the VNs \(x_i\)  of the FG, for \(i \in \lbrace 1,...,N\rbrace \) .
The aim of nesting an NN to the FG is two-fold:
